Invisible Control of Self-Organizing Agents Leaving Unknown Environments

Transcription

1 Invisible Control of Self-Organizing Agents Leaving Unknown Environments (joint work with G. Albi, E. Cristiani, and D. Kalise) Mattia Bongini Technische Universität München, Department of Mathematics, Chair of Applied Numerical Analysis OCERTO 2016 Cortona June 20-24, 2016 Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 1 of 22

2 Pedestrian dynamics: multiscale models and control Rising interest towards modeling and control of large groups of individuals: Models for pedestrian dynamics: Hughes 02, Maury-Chupin-Santambrogio 11, Di Francesco-Markowich-Pietschmann-Wolfram 11; Control of multiagent systems via external agents: B.-Buttazzo 16 (theoretical), Albi-Pareschi-Zanella 14, Pinnau-Totzeck-Tse-Burger 16 (numerical), Butail-Bartolini-Porfiri 13 (practical); Multiscale modeling and mean-field optimal control: Fornasier-Piccoli-Rossi 14, B.-Fornasier-Rossi-Solombrino 15, Cristiani-Piccoli-Tosin 11; List is far from exhaustive, but shows two central issues: dimensionality reduction of micro models via mesoscopic ones when number of agents explodes; influencing the behavior of the crowd via controlled external agents. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 2 of 22

3 Pedestrian dynamics: multiscale models and control Rising interest towards modeling and control of large groups of individuals: Models for pedestrian dynamics: Hughes 02, Maury-Chupin-Santambrogio 11, Di Francesco-Markowich-Pietschmann-Wolfram 11; Control of multiagent systems via external agents: B.-Buttazzo 16 (theoretical), Albi-Pareschi-Zanella 14, Pinnau-Totzeck-Tse-Burger 16 (numerical), Butail-Bartolini-Porfiri 13 (practical); Multiscale modeling and mean-field optimal control: Fornasier-Piccoli-Rossi 14, B.-Fornasier-Rossi-Solombrino 15, Cristiani-Piccoli-Tosin 11; List is far from exhaustive, but shows two central issues: dimensionality reduction of micro models via mesoscopic ones when number of agents explodes; influencing the behavior of the crowd via controlled external agents. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 2 of 22

4 Pedestrian dynamics: multiscale models and control Rising interest towards modeling and control of large groups of individuals: Models for pedestrian dynamics: Hughes 02, Maury-Chupin-Santambrogio 11, Di Francesco-Markowich-Pietschmann-Wolfram 11; Control of multiagent systems via external agents: B.-Buttazzo 16 (theoretical), Albi-Pareschi-Zanella 14, Pinnau-Totzeck-Tse-Burger 16 (numerical), Butail-Bartolini-Porfiri 13 (practical); Multiscale modeling and mean-field optimal control: Fornasier-Piccoli-Rossi 14, B.-Fornasier-Rossi-Solombrino 15, Cristiani-Piccoli-Tosin 11; List is far from exhaustive, but shows two central issues: dimensionality reduction of micro models via mesoscopic ones when number of agents explodes; influencing the behavior of the crowd via controlled external agents. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 2 of 22

5 Pedestrian dynamics: multiscale models and control Rising interest towards modeling and control of large groups of individuals: Models for pedestrian dynamics: Hughes 02, Maury-Chupin-Santambrogio 11, Di Francesco-Markowich-Pietschmann-Wolfram 11; Control of multiagent systems via external agents: B.-Buttazzo 16 (theoretical), Albi-Pareschi-Zanella 14, Pinnau-Totzeck-Tse-Burger 16 (numerical), Butail-Bartolini-Porfiri 13 (practical); Multiscale modeling and mean-field optimal control: Fornasier-Piccoli-Rossi 14, B.-Fornasier-Rossi-Solombrino 15, Cristiani-Piccoli-Tosin 11; List is far from exhaustive, but shows two central issues: dimensionality reduction of micro models via mesoscopic ones when number of agents explodes; influencing the behavior of the crowd via controlled external agents. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 2 of 22

6 Pedestrian dynamics: multiscale models and control Rising interest towards modeling and control of large groups of individuals: Models for pedestrian dynamics: Hughes 02, Maury-Chupin-Santambrogio 11, Di Francesco-Markowich-Pietschmann-Wolfram 11; Control of multiagent systems via external agents: B.-Buttazzo 16 (theoretical), Albi-Pareschi-Zanella 14, Pinnau-Totzeck-Tse-Burger 16 (numerical), Butail-Bartolini-Porfiri 13 (practical); Multiscale modeling and mean-field optimal control: Fornasier-Piccoli-Rossi 14, B.-Fornasier-Rossi-Solombrino 15, Cristiani-Piccoli-Tosin 11; List is far from exhaustive, but shows two central issues: dimensionality reduction of micro models via mesoscopic ones when number of agents explodes; influencing the behavior of the crowd via controlled external agents. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 2 of 22

7 Pedestrian dynamics: multiscale models and control Rising interest towards modeling and control of large groups of individuals: Models for pedestrian dynamics: Hughes 02, Maury-Chupin-Santambrogio 11, Di Francesco-Markowich-Pietschmann-Wolfram 11; Control of multiagent systems via external agents: B.-Buttazzo 16 (theoretical), Albi-Pareschi-Zanella 14, Pinnau-Totzeck-Tse-Burger 16 (numerical), Butail-Bartolini-Porfiri 13 (practical); Multiscale modeling and mean-field optimal control: Fornasier-Piccoli-Rossi 14, B.-Fornasier-Rossi-Solombrino 15, Cristiani-Piccoli-Tosin 11; List is far from exhaustive, but shows two central issues: dimensionality reduction of micro models via mesoscopic ones when number of agents explodes; influencing the behavior of the crowd via controlled external agents. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 2 of 22

8 Pedestrian dynamics: multiscale models and control Rising interest towards modeling and control of large groups of individuals: Models for pedestrian dynamics: Hughes 02, Maury-Chupin-Santambrogio 11, Di Francesco-Markowich-Pietschmann-Wolfram 11; Control of multiagent systems via external agents: B.-Buttazzo 16 (theoretical), Albi-Pareschi-Zanella 14, Pinnau-Totzeck-Tse-Burger 16 (numerical), Butail-Bartolini-Porfiri 13 (practical); Multiscale modeling and mean-field optimal control: Fornasier-Piccoli-Rossi 14, B.-Fornasier-Rossi-Solombrino 15, Cristiani-Piccoli-Tosin 11; List is far from exhaustive, but shows two central issues: dimensionality reduction of micro models via mesoscopic ones when number of agents explodes; influencing the behavior of the crowd via controlled external agents. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 2 of 22

9 The evacuation problem The crowd find its way to the exit thanks to the two fuchsia leaders. Our goal is to evacuate a crowd of individuals from an environment they don t know under limited visibility. We show that invisible sparse strategies (i.e., by means of few, unrecognized, external agents) influence the crowd effectively and improve evacuation. Through Boltzmann equation we propose a mesoscopic description of this dynamics when the number of pedestrians is large. Develop a set of numerical techniques for computing optimal exit strategies in both cases. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 3 of 22

10 The evacuation problem The crowd find its way to the exit thanks to the two fuchsia leaders. Our goal is to evacuate a crowd of individuals from an environment they don t know under limited visibility. We show that invisible sparse strategies (i.e., by means of few, unrecognized, external agents) influence the crowd effectively and improve evacuation. Through Boltzmann equation we propose a mesoscopic description of this dynamics when the number of pedestrians is large. Develop a set of numerical techniques for computing optimal exit strategies in both cases. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 3 of 22

11 The evacuation problem The crowd find its way to the exit thanks to the two fuchsia leaders. Our goal is to evacuate a crowd of individuals from an environment they don t know under limited visibility. We show that invisible sparse strategies (i.e., by means of few, unrecognized, external agents) influence the crowd effectively and improve evacuation. Through Boltzmann equation we propose a mesoscopic description of this dynamics when the number of pedestrians is large. Develop a set of numerical techniques for computing optimal exit strategies in both cases. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 3 of 22

12 The evacuation problem The crowd find its way to the exit thanks to the two fuchsia leaders. Our goal is to evacuate a crowd of individuals from an environment they don t know under limited visibility. We show that invisible sparse strategies (i.e., by means of few, unrecognized, external agents) influence the crowd effectively and improve evacuation. Through Boltzmann equation we propose a mesoscopic description of this dynamics when the number of pedestrians is large. Develop a set of numerical techniques for computing optimal exit strategies in both cases. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 3 of 22

13 Model guidelines: followers (i.e., evacuees) The non-controlled agents of the crowd are called followers, and are subject to a second-order dynamics with an isotropic metric short-range repulsion force; a relaxation term toward a given characteristic speed; if the exit is not visible an isotropic topological alignment force, i.e., given N N, the i-th agent aligns with those inside B N (x i, t), the smallest ball at time t containing at least N agents. a random walk, in order to explore the unknown environment; if the exit is visible a sharp motion toward the exit. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 4 of 22

14 Model guidelines: followers (i.e., evacuees) The non-controlled agents of the crowd are called followers, and are subject to a second-order dynamics with an isotropic metric short-range repulsion force; a relaxation term toward a given characteristic speed; if the exit is not visible an isotropic topological alignment force, i.e., given N N, the i-th agent aligns with those inside B N (x i, t), the smallest ball at time t containing at least N agents. a random walk, in order to explore the unknown environment; if the exit is visible a sharp motion toward the exit. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 4 of 22

15 Model guidelines: followers (i.e., evacuees) The non-controlled agents of the crowd are called followers, and are subject to a second-order dynamics with an isotropic metric short-range repulsion force; a relaxation term toward a given characteristic speed; if the exit is not visible an isotropic topological alignment force, i.e., given N N, the i-th agent aligns with those inside B N (x i, t), the smallest ball at time t containing at least N agents. a random walk, in order to explore the unknown environment; if the exit is visible a sharp motion toward the exit. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 4 of 22

16 Model guidelines: followers (i.e., evacuees) The non-controlled agents of the crowd are called followers, and are subject to a second-order dynamics with an isotropic metric short-range repulsion force; a relaxation term toward a given characteristic speed; if the exit is not visible an isotropic topological alignment force, i.e., given N N, the i-th agent aligns with those inside B N (x i, t), the smallest ball at time t containing at least N agents. a random walk, in order to explore the unknown environment; if the exit is visible a sharp motion toward the exit. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 4 of 22

17 Model guidelines: followers (i.e., evacuees) The non-controlled agents of the crowd are called followers, and are subject to a second-order dynamics with an isotropic metric short-range repulsion force; a relaxation term toward a given characteristic speed; if the exit is not visible an isotropic topological alignment force, i.e., given N N, the i-th agent aligns with those inside B N (x i, t), the smallest ball at time t containing at least N agents. a random walk, in order to explore the unknown environment; if the exit is visible a sharp motion toward the exit. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 4 of 22

18 Model guidelines: followers (i.e., evacuees) The non-controlled agents of the crowd are called followers, and are subject to a second-order dynamics with an isotropic metric short-range repulsion force; a relaxation term toward a given characteristic speed; if the exit is not visible an isotropic topological alignment force, i.e., given N N, the i-th agent aligns with those inside B N (x i, t), the smallest ball at time t containing at least N agents. a random walk, in order to explore the unknown environment; if the exit is visible a sharp motion toward the exit. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 4 of 22

19 Model guidelines: leaders (i.e., stewards) The controlled agents of the crowd are called leaders. They are less than followers (N L N F ) and evolve according to a first-order dynamics with an isotropic metric short-range repulsion force; an optimal force which is the result of an offline optimization procedure, minimizing some cost functional. First vs. second-order model: for followers a second-order model is necessary since they must perceive velocities to align. The bigger inertia is compensated by stronger forces w.r.t. the ones in leaders dynamics. Metric vs. topological interaction: alignment is topological since empirical evidence suggests that only close neighbors play a role, (see Ballerini et al. 08). Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 5 of 22

20 Model guidelines: leaders (i.e., stewards) The controlled agents of the crowd are called leaders. They are less than followers (N L N F ) and evolve according to a first-order dynamics with an isotropic metric short-range repulsion force; an optimal force which is the result of an offline optimization procedure, minimizing some cost functional. First vs. second-order model: for followers a second-order model is necessary since they must perceive velocities to align. The bigger inertia is compensated by stronger forces w.r.t. the ones in leaders dynamics. Metric vs. topological interaction: alignment is topological since empirical evidence suggests that only close neighbors play a role, (see Ballerini et al. 08). Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 5 of 22

21 Model guidelines: leaders (i.e., stewards) The controlled agents of the crowd are called leaders. They are less than followers (N L N F ) and evolve according to a first-order dynamics with an isotropic metric short-range repulsion force; an optimal force which is the result of an offline optimization procedure, minimizing some cost functional. First vs. second-order model: for followers a second-order model is necessary since they must perceive velocities to align. The bigger inertia is compensated by stronger forces w.r.t. the ones in leaders dynamics. Metric vs. topological interaction: alignment is topological since empirical evidence suggests that only close neighbors play a role, (see Ballerini et al. 08). Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 5 of 22

22 Model guidelines: leaders (i.e., stewards) The controlled agents of the crowd are called leaders. They are less than followers (N L N F ) and evolve according to a first-order dynamics with an isotropic metric short-range repulsion force; an optimal force which is the result of an offline optimization procedure, minimizing some cost functional. First vs. second-order model: for followers a second-order model is necessary since they must perceive velocities to align. The bigger inertia is compensated by stronger forces w.r.t. the ones in leaders dynamics. Metric vs. topological interaction: alignment is topological since empirical evidence suggests that only close neighbors play a role, (see Ballerini et al. 08). Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 5 of 22

23 Microscopic model For i = 1,..., N F and k = 1,..., N L ẋ i = v i, v i ẏ k = A(x i, v i) + N F j=1 H(xi, vi, xj, vj) + N L l=1 H(xi, vi, y l, w l ) = w k = N F j=1 R ζ,r(y k, x j) + N L l=1 R ζ,r(y k, y l ) + u k, Let θ(x) represents the characteristic function of the target s visibility zone and ( ) A(x, v) := (1 θ(x))c z (z v)+θ(x)c d x d x x d x v +C v (α 2 v 2 )v, where z N (0, σ 2 ), α is the characteristic speed. H(x, v, y, w) := C r FR γ,r(x, y) + (1 θ(x)) Ca N (w v) χ B N (x,t)(y) R γ,r(x, y) = In the dynamics of y k, ζ γ. { e y x γ y x if y B y x r(x)\{x}, 0 otherwise; Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 6 of 22

24 Microscopic model For i = 1,..., N F and k = 1,..., N L ẋ i = v i, v i ẏ k = A(x i, v i) + N F j=1 H(xi, vi, xj, vj) + N L l=1 H(xi, vi, y l, w l ) = w k = N F j=1 R ζ,r(y k, x j) + N L l=1 R ζ,r(y k, y l ) + u k, Let θ(x) represents the characteristic function of the target s visibility zone and ( ) A(x, v) := (1 θ(x))c z (z v)+θ(x)c d x d x x d x v +C v (α 2 v 2 )v, where z N (0, σ 2 ), α is the characteristic speed. H(x, v, y, w) := C r FR γ,r(x, y) + (1 θ(x)) Ca N (w v) χ B N (x,t)(y) R γ,r(x, y) = In the dynamics of y k, ζ γ. { e y x γ y x if y B y x r(x)\{x}, 0 otherwise; Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 6 of 22

25 Microscopic model For i = 1,..., N F and k = 1,..., N L ẋ i = v i, v i ẏ k = A(x i, v i) + N F j=1 H(xi, vi, xj, vj) + N L l=1 H(xi, vi, y l, w l ) = w k = N F j=1 R ζ,r(y k, x j) + N L l=1 R ζ,r(y k, y l ) + u k, Let θ(x) represents the characteristic function of the target s visibility zone and ( ) A(x, v) := (1 θ(x))c z (z v)+θ(x)c d x d x x d x v +C v (α 2 v 2 )v, where z N (0, σ 2 ), α is the characteristic speed. H(x, v, y, w) := C r FR γ,r(x, y) + (1 θ(x)) Ca N (w v) χ B N (x,t)(y) R γ,r(x, y) = In the dynamics of y k, ζ γ. { e y x γ y x if y B y x r(x)\{x}, 0 otherwise; Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 6 of 22

26 Microscopic model For i = 1,..., N F and k = 1,..., N L ẋ i = v i, v i ẏ k = A(x i, v i) + N F j=1 H(xi, vi, xj, vj) + N L l=1 H(xi, vi, y l, w l ) = w k = N F j=1 R ζ,r(y k, x j) + N L l=1 R ζ,r(y k, y l ) + u k, Let θ(x) represents the characteristic function of the target s visibility zone and ( ) A(x, v) := (1 θ(x))c z (z v)+θ(x)c d x d x x d x v +C v (α 2 v 2 )v, where z N (0, σ 2 ), α is the characteristic speed. H(x, v, y, w) := C r FR γ,r(x, y) + (1 θ(x)) Ca N (w v) χ B N (x,t)(y) R γ,r(x, y) = In the dynamics of y k, ζ γ. { e y x γ y x if y B y x r(x)\{x}, 0 otherwise; Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 6 of 22

27 Dynamics of followers Followers dynamics without leaders Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 7 of 22

28 Dynamics of followers Followers dynamics without leaders If not influenced, the random term + topo- Influence of external agents with preferred logical alignment splits the followers direction: when removed, the group splits Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 7 of 22

29 Control strategy: dumb The control u k : [0, T ] R d, k = 1,..., N L as dumb strategy: ( ) x d y k (t) u k (t) = x d y k (t) y k(t) (Left) Dynamics with dumb strategy and (Right) occupancy of the exit s visibility zone Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 8 of 22

30 Smart strategy: Modified Compass Search The control u k : [0, T ] R d, k = 1,..., N L, u k minimizes the functional ( T N L J(u) = P (t) + ν u k (t) )dt, 2 0 k=1 Where P (t) is the number of followers outside exit at time t. We proceed via Modified Compass Search 1 : leaders trajectories are piecewise constant; starting from an initial guess, at each iteration we modify the current best strategy with small random variations; we keep the variation if the evaluated cost is smaller than before; the method generates a sequence converging to a local minimum. 1 Audet-Dang-Orban 14 Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 9 of 22

31 Clog up effect around exit Dynamics with dumb strategy Dynamics with smart strategy Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 10 of 22

32 Clog up effect around exit Dynamics with dumb strategy Dynamics with smart strategy Occupancy of the exit s visibility zone with dumb strategy Occupancy of the exit s visibility zone with smart strategy Good strategies avoid exit s clog up, hence congestion drop. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 10 of 22

33 Mean-field approximation The number of interacting agents can be rather large, thus we need to solve a very large system of ODEs, which can constitute a serious difficulty. A natural way to tackle this problem is to approximate the problem with a kinetic equation. The problem of rigorously derive a mean-field equation from a system of interacting particles in the case of swarming and flocking models is well studied 2 also for an optimal control problem 3. We consider here binary interaction Boltzmann-type and mean-field approximation of the microscopic dynamic 4. This approach is related with the development of Boltzmann collision algorithms in plasma physics 5 2 Canizo-Carrillo-Rosado 10, Carrillo-Y. P. Choi-Hauray-Salem 15 3 Fornasier-Solombrino 13, Fornasier-Piccoli-Rossi 14, B.-Fornasier-Rossi-Solombrino 15 4 Carrillo-Fornasier-Toscani-Vecil 10, Pareschi-Toscani 13 5 Bird 94, Bobylev-Nanbu 00, Caflisch-Pareschi-Dimarco 10 Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 11 of 22

34 Boltzmann approximation Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 12 of 22

35 Mean-field controlled dynamic The mean-field limit of the microscopic model describes the evolution of the continuous density of followers f = f(t, x, v) coupled with the evolution of the microscopic leaders tf + v xf = v ((S + H[f] + H[g])f) σ2 (θc z ) 2 vf, N L ẏ k = R ζ,r (y k, x)f(x, v) dx dv + R ζ,r (y k, y l ) + u k, R 2d l=1 k = 1,..., N L Where the non-linear interactions terms are H[f](x, v) = H(x, ˆx, v, ˆv)f(ˆx, ˆv) dˆx dˆv, R 2d S(x, v) = θ(x)c z v + (1 θ(x))c d ( x d x x d x v (MF) ) + C v (α 2 v 2 )v. Density g = g(t, x, v) = N L k=1 δ y k (t),w k (t)(x, v), represents the empirical measure of leaders and u k is given by the solution of an optimal control problem. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 13 of 22

36 Mean-field controlled dynamic The mean-field limit of the microscopic model describes the evolution of the continuous density of followers f = f(t, x, v) coupled with the evolution of the microscopic leaders tf + v xf = v ((S + H[f] + H[g])f) σ2 (θc z ) 2 vf, N L ẏ k = R ζ,r (y k, x)f(x, v) dx dv + R ζ,r (y k, y l ) + u k, R 2d l=1 k = 1,..., N L Where the non-linear interactions terms are H[f](x, v) = H(x, ˆx, v, ˆv)f(ˆx, ˆv) dˆx dˆv, R 2d S(x, v) = θ(x)c z v + (1 θ(x))c d ( x d x x d x v (MF) ) + C v (α 2 v 2 )v. Density g = g(t, x, v) = N L k=1 δ y k (t),w k (t)(x, v), represents the empirical measure of leaders and u k is given by the solution of an optimal control problem. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 13 of 22

37 Kinetic approximation We assume that f, g satisfy the following Boltzmann-Povzner dynamics 6 + the ODEs of leaders tf + v xf = λ F Q F (f, f) + λ L Q L (f, g) N L ẏ k = R ζ,r (y k, x)f(x, v) dx dv + R ζ,r (y k, y l ) + u k, R 2d l=1 k = 1,..., N L (BP) where λ F and λ L are the interaction frequencies and [ ( ) Q F 1 (f, f) = E f(x, v )f(ˆx, ˆv ) f(x, v)f(ˆx, ˆv) R 2d J FF [ ( ) Q L 1 (f, g) = E f(x, v )g( x, ṽ ) f(x, v)g( x, ṽ) J FL R 2d ] dˆx dˆv, ] d x dṽ. with v, v indicates the pre-interaction velocities and E[ ] is the expected value w.r.t. ξ. 6 Povzner 62 Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 14 of 22

38 Kinetic approximation We assume that f, g satisfy the following Boltzmann-Povzner dynamics 6 + the ODEs of leaders tf + v xf = λ F Q F (f, f) + λ L Q L (f, g) N L ẏ k = R ζ,r (y k, x)f(x, v) dx dv + R ζ,r (y k, y l ) + u k, R 2d l=1 k = 1,..., N L (BP) where λ F and λ L are the interaction frequencies and [ ( ) Q F 1 (f, f) = E f(x, v )f(ˆx, ˆv ) f(x, v)f(ˆx, ˆv) R 2d J FF [ ( ) Q L 1 (f, g) = E f(x, v )g( x, ṽ ) f(x, v)g( x, ṽ) J FL R 2d ] dˆx dˆv, ] d x dṽ. with v, v indicates the pre-interaction velocities and E[ ] is the expected value w.r.t. ξ. 6 Povzner 62 Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 14 of 22

39 Binary interactions 7 When a follower (x, v) interacts with another follower (ˆx, ˆv), they update their state variables according to v = v + η F [θ(x)c z ξ + S(x, v) +N F H(x, v, ˆx, ˆv)] }{{} (FF) A(x,v) ˆv = ˆv + η [ F θ(ˆx)c z ξ + S(ˆx, ˆv) + N F H(ˆx, ˆv, x, v) ] where η F is the strength of the interaction follower-follower, and ξ N (0, ς 2 ). When a follower (x, v) interacts with a leader ( x, ṽ), (FL) { v = v + η L N L H(x, v, x, ṽ) ṽ = ṽ where η L is the strength of the interaction follower-leader. 7 Cercignani-Illner-Pulvirenti 94 Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 15 of 22

40 Theorem: Grazing collision limit Suppose that for some δ (0, 1) ( E ξ 2+δ) is finite, the functions S and H are in L p loc for p = 2, 2 + δ. Fix the control u. Let consider the following scaling η F = η L = ε, λ F =λ L = 1 ε, ς2 = σ2 ε and define (f ε, y ε ) be a solution of (BP). Then, as ε 0, (f ε, y ε ) converges pointwise to a solution of (MF), the mean-field limit of the microscopic model. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 16 of 22

41 Idea of the proof 8 Let T be a function space with suitably smooth test functions. Then for ϕ T δ, the weak formulation of (BP) reads ( ) λ Q(f, f), ϕ = λe (ϕ(x, v ) ϕ(x, v)) f(x, v)f(ˆx, ˆv) dxdvdˆxdˆv R 4d. We derive the Taylor expansion around v v up to the second order of the weak interaction operator, obtaining ( λ Q(f, f), ϕ = λe vϕ(x, v) (v v)f(x, v)f(ˆx, ˆv) dxdvdˆxdˆv+ R 4d + λ ) d v (i,j) ϕ(x, v) (v v) 2 i (v v) j f(x, v)f(ˆx, ˆv) dx dv dˆx dˆv + λr ϕ R 4d i,j=1 Thanks to the boundedness of S and H we have that the remainder R ϕ is bounded. Therefore using the grazing collision scaling, and taking the limit for ε 0 for any ϕ T δ we obtain the mean-field equation (MF). 8 Toscani 06 Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 17 of 22

42 Binary Interaction algorithm The main idea is to simulate the binary interaction dynamic for small values of ε in order to approximate the Fokker-Planck equation model 9. We use a splitting method, for transport and collisional part of the scaled Boltzmann equation (BP). tf = v xf tf = 1 ε QF ε (f, f) + 1 ε QL ε (f, g) (T) (C) In order to simulate the evolution of (C) we use a Monte-Carlo method. The algorithms cost is linear, O(N s), w.r.t. to the number of sample particles N s used to reconstruct the density f. The resulting algorithm is fully meshless since the binary interactions are non local. 9 Albi-Pareschi 13 Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 18 of 22

43 Uncontrolled case Evacuated Mass Mass inside Σ Time steps Uncontrolled evolution of followers density. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 19 of 22

44 Dumb strategy Control u : [0, T ] R d as a dumb strategy: u k (t) = ( ) x d yk (t) y x d y k (t) k(t) ; Evacuated Mass Mass inside Σ Time steps Evolution of followers density under microscopic leaders fixed strategy. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 20 of 22

45 Smart strategy Control u : [0, T ] R d as a smart strategy such that ( T min J(u) = P (t) + ν u 0 u k (t) )dt, 2 N L k=1 where P (t) represents the mass of followers outside exit at time t Evacuated Mass Mass inside Σ Time steps Evolution of followers density under microscopic leaders smart strategy. Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 21 of 22

46 A few info WWW: References: G. Albi, M. Bongini, E. Cristiani, and D. Kalise, Invisible control of self-organizing agents leaving unknown environments, to apper in SIAM Journal on Applied Mathematics, 2016 M. Bongini and G. Buttazzo, Optimal control problems in transport dynamics, submitted, 2016 M. Bongini, M. Fornasier, F. Rossi, and F. Solombrino, Mean-field Pontryagin maximum principle, submitted, 2016 G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Modeling & Simulation, pp. 1 29, Volume 11, Issue 1, 2013 Mattia Bongini Invisible Control of Agents Leaving Unknown Environments 22 of 22